A new method for solving fully fuzzy linear programming problems

Annals of Fuzzy Mathematics and Informatics Volume 3, No. 1, (January 2012), pp. 103- 118 ISSN 2093–9310 http://www.afmi.or.kr

@FMI c Kyung Moon Sa Co.

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A new method for solving fully fuzzy linear programming problems Amit Kumar, Pushpinder Singh Received 22 April 2011; Accepted 4 July 2011

Abstract. Several authors have used ranking function for solving fuzzy linear programming problems. In this paper some fuzzy linear programming problems are chosen which can’t be solved by using any of the existing methods and a new method is proposed to solve such type of fuzzy linear programming problems. The main advantage of the proposed method over existing methods is that the fuzzy linear programming problems which can be solved by the existing methods can also be solved by the proposed method but there exist several fuzzy linear programming problems which can be solved only by using the proposed method i.e., it is not possible to solve these fuzzy linear programming problems by using the existing methods. To show the advantage of the proposed method over existing methods some fuzzy linear programming problems, which can’t be solved by the existing methods, are solved by the proposed method.

2010 AMS Classification: 03E72, 90C70 Keywords: Fuzzy linear programming problems, Ranking function, L-R flat fuzzy numbers. Corresponding Author: Pushpinder Singh ([email protected])

1. Introduction

L

inear programming is one of the most frequently applied operations research techniques. Although it is investigated and expanded for more than six decades by many researchers from the various point of views, it is still useful to develop new approaches in order to better fit the real world problems within the framework of linear programming. Any linear programming model representing real world situations involves a lot of parameters whose values are assigned by experts, and in the conventional approach, they are required to fix an exact value to the aforementioned parameters. However, both experts and the decision makers frequently do not precisely know the value of those parameters. If exact values are suggested these are

Amit Kumar et al./Ann. Fuzzy Math. Inform. 3 (2012), No. 1, 103–118

only statistical inference from past data and their stability is doubtful, so the parameters of the problem are usually defined by the decision makers in an uncertain way or by means of language statement parameters. Therefore, it is useful to consider the knowledge of experts about the parameters as fuzzy data [21]. The concept of fuzzy mathematical programming on general level was first proposed by Tanaka et al. [18] in the framework of the fuzzy decision of Bellman and Zadeh [2]. The first formulation of fuzzy linear programming (FLP) problems was proposed by Zimmermann [20]. Afterwards, many authors [3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17] considered various types of the FLP problems and proposed several approaches for solving these problems. Real numbers can be linearly ordered by the relation ≥ or ≤, however this type of inequality does not exist in fuzzy numbers. Since fuzzy numbers are represented by possibility distribution, they can overlap with each other and it is difficult to determine clearly whether one fuzzy number is larger or smaller than other. An efficient method for ordering the fuzzy numbers is by the use of a ranking function, which maps each fuzzy number into the real line, where a natural order exists. Jain [9] proposed the concept of ranking function for comparing normal fuzzy numbers. Maleki [14] proposed a method to unify some of the existing approaches which are using different ranking functions for solving fuzzy programming problems. Moreover, they introduced a method for solving linear programming with vagueness in constraints by using linear ranking function. Maleki et al. [16] proposed method for solving fuzzy number linear programming problems using the concept of ranking of fuzzy numbers. Nasseri and Ardil [17] developed simplex method to FLP problems by using ceratin ranking function. This method uses simplex tableau which is used for solving linear programming problems in crisp environment before. Mahadavi and Nasseri [12, 13] proposed duality results and a dual simplex method to solve FLP problems with trapezoidal fuzzy variables by the use of ranking function. Ganesan and Veeramani [7] dealt with a kind of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. Allahviranloo et al. [1] proposed a new method for solving fully fuzzy linear programming problem by using ranking function taking variables as restricted fuzzy numbers. In this paper some fuzzy linear programming problems are chosen which can’t be solved by using any of the existing methods and a new method is proposed to solve such type of fuzzy linear programming problems. The main advantage of the proposed method over existing methods is that the fuzzy linear programming problems which can be solved by the existing methods can also be solved by the proposed method but there are several fuzzy linear programming problems which can be solved only by using the proposed method i.e., it is not possible to solve these fuzzy linear programming problems by using the existing methods. To show the advantage of the proposed method over existing methods some fuzzy linear programming problems, which can’t be solved by using the existing methods, are solved by the proposed method. This paper is organized as follows: In Section 2, some basic definitions, arithmetic operations and Yager’s ranking approach [19] for the ranking of fuzzy numbers are reviewed. In Section 3 formulation of FLP problem is discussed. In Section 4, the shortcomings of existing method are pointed out. In Section 5, modified formulation of FLP problems is proposed. In Section 6, applicability of existing method is discussed. In Section 7, the limitations of 104


existing method are pointed out. In Section 8, a new method is proposed to find the optimal solution of FLP problems. In Section 9, advantages of proposed methods are discussed. In Section 10, results obtained by the existing methods and by proposed method are compared. The conclusion is discussed in Section 11. 2. Preliminaries In this section some basic definitions, arithmetic operations and Yager’s ranking function approach for the ranking of fuzzy numbers are reviewed. 2.1. Basic definitions. In this section some basic definitions are reviewed [6]. e = (m, n, α, β)LR is said to be an L-R flat fuzzy Definition 2.1. A fuzzy number A number if m−x for x ≤ m, α > 0  L( α ), R( x−n ), for x ≥ n, β > 0 µAe(x) = β  1, otherwise e = (m, n, α, β)LR will be converted into A e = (m, α, β)LR and is If m = n then A said to be an L-R fuzzy number. L and R are called reference functions, which are continuous, non-increasing functions that defining the left and right shapes of µAe(x) respectively and L(0) = R(0) = 1. Two special cases are triangular and trapezoidal fuzzy number, for which L(x) = R(x) = maximum {0, 1 − x}, are linear functions. Three commonly used nonlinear reference functions with parameters q, denoted as RFq , are summarized as follows: power: RFq (x) = maximum (0, 1 − xq ), q ≥ 0, q exponential power: RFq (x) = e−x , q ≥ 0, 1 rational: RFq (x) = (1+x q ≥ 0. q) , e = (m, n, α, β)LR be an L-R flat fuzzy number and λ be a Definition 2.2. Let A real number in the interval [0, 1] then the crisp set Aλ = {x ∈ X : µA˜ (x) ≥ λ} = [m − αL−1 (λ), n + βR−1 (λ)], e is said to be λ-cut of A. Definition 2.3. A L-R flat fuzzy number A˜ = (m, n, α, β)LR is said to be nonnegative L-R flat fuzzy number if m − α ≥ 0. 2.2. Arithmetic operations. Let A˜1 = (m1 , n1 , α, β)LR be non-negative L-R flat fuzzy numbers and A˜2 = (m2 , n2 , α0 , β 0 )LR , A˜3 = (m3 , n3 , α00 , β 00 )LR be any L-R flat fuzzy numbers then (i) A˜1 ⊗ A˜2 = (m1 m2 , n1 n2 , m1 α0 + m2 α, n1 β 0 + n2 β)LR for m1 − α > 0 and m2 − α 0 > 0 0 00 0 00 (ii) A˜2 ⊕ A˜ 3 = (m2 + m3 , n2 + n3 , α + α , β + β )LR 0 0 λ ≥ 0; (λm2 , λn2 , λα , λβ )LR , (iii) λA˜2 = (λm2 , λn2 , −λβ 0 , −λα0 )RL , λ ≤ 0. (iv) A˜1 ⊗ A˜2 = (a1 , a2 , a3 , a4 )LR , 105


where, a1 = minimum (m1 m2 , n1 n2 ), a2 = maximum (m1 n2 , n1 n2 ), a3 = minimum (m1 m2 , n1 m2 ) − minimum ((m1 − α)(m2 − α0 ), (β + n1 )(m2 − α0 )), a4 = maximum (m1 n2 , n1 n2 ) − maximum((m1 − α)(β 0 + n2 ), (β + n1 )(β 0 + n2 ))) and minimum ((m1 − α)(β 0 − n2 ), (β + n1 )(m2 − α0 )) = −| (m1 −α)(β

0

+n2 )−(β+n1 )(m2 −α0 ) | 2

and maximum ((β + n1 )(β 0 + n2 ), (m1 − α)(m2 − α0 )) = +| (β+n1 )(β

0

0

(m1 −α)(β 0 +n2 )+(β+n1 )(m2 −α0 ) 2

(β+n1 )(β 0 +n2 )+(m1 −α)(m2 −α0 ) 2

+n2 )−(m1 −α)(m2 −α ) | 2

2.3. Yager’s ranking approach. A number of approaches have been proposed for the ranking of fuzzy numbers. A relatively simple computational and easily understandable ranking method proposed by Yager [19] is considered for the ranking of fuzzy numbers in this paper. Yager [19] proposed a procedure for ordering fuzzy sets ˜ is calculated for the fuzzy number A e = (m, n, α, β)LR in which a ranking index